Uncertainty and Growth Disasters

K.7 Uncertainty and Growth Disasters Jovanovic, Boyan and Sai Ma Please cite paper as: Boyan, Jovanovic and Sai Ma (2020). Uncertainty and Growth Disasters. International Finance Discussion Papers 1279. https://doi.org/10.17016/IFDP.2020.1279 International Finance Discussion Papers Board of Governors of the Federal Reserve System Number 1279 May 2020

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 1279 May 2020 Uncertainty and Growth Disasters Boyan Jovanovic and Sai Ma NOTE: International Finance Discussion Papers (IFDPs) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the International Finance Discussion Papers Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at www.ssrn.com.

Uncertainty and Growth Disasters Boyan Jovanovic Sai Ma (cid:3) New York University Federal Reserve Board March 2020 Abstract This paper documents several stylized facts on the real e⁄ects of economic uncertainty. First, higher uncertainty is associated with a more dispersed and negatively skewed distribution of output growth. Second, the response of economic growth to an increase in uncertainty is highly nonlinear and asymmetric. Third, higher asset volatility magni(cid:133)es the negative impact of uncertainty on growth. We develop and estimate an analytically tractablemodelinwhichrapidadoptionofnewtechnologymayraiseeconomicuncertainty which causes measured productivity to decline. The equilibrium growth distribution is negatively skewed and higher uncertainty leads to a thicker left tail. JEL Code: D80, E44, O40, O47 Keywords: Uncertainty and growth, volatility, downside risk, growth at risk Jovanovic is with Department of Economics, New York University (boyan.jovanovic@nyu.edu). Ma (cid:3) is with the Board of Governors of the Federal Reserve System, Division of International Finance (Corresponding author). C. Ave & 20th Street NW, Washington, DC 20551. Email: sai.ma@frb.gov. The views expressed are those of the authors and not necessarily those of the Federal Reserve Board or the Federal Reserve System. All errors are ours. First draft: May 2019.

1 Introduction Contemporary macro literature often (cid:133)nds uncertainty about the future to be an important driver of economic (cid:135)uctuations. A growing body of work proposes uncertainty as a cause of economic slowdown and sluggish recovery. For example, Bloom (2009) and Bloom, Floetotto, Jaimovich, Saporta-Eksten and Terry (2018) argue that higher uncertainty stems from the process governing technological innovation, which subsequently causes a decline in real activity. Empirically, this evidence has been found to be robust to the use of various proxy variables such as implied stock volatility (VIX), economic policy uncertainty (EPU) from Baker, BloomandDavis(2016), orabroad-basedmeasureofmacroeconomicand(cid:133)nancial uncertainty, as in Jurado, Ludvigson and Ng (2015) (JLN) and Ludvigson, Ma and Ng (2019) (LMN). However, these papers usuallyinvestigate the impact of higheruncertainty on expected mean growth either via a linear forecasting regression or a structural vector autoregression(SVAR).Consequently,theyaresilentaboutthee⁄ectofuncertaintyonthe volatilityorother higher momentsofthegrowthdistribution, andthismayunderestimate the impact of economic uncertainty on growth downside risk. Inthispaper, weprovideevidencethatuncertaintyishighlycorrelatedwiththehigher moments of the growth distribution. Figure 1 depicts the contemporaneous relationship between JLN macro uncertainty and 36-month rolling window average industrial production(IP)growth(leftpanel), growthvolatility(middlepanel), andgrowthskewness(right panel). While uncertainty is highly negatively correlated ( 36%) with the mean growth (cid:0) as shown in the literature, we (cid:133)nd that uncertainty is also highly correlated with growth volatility (44%) and growth skewness ( 22%), respectively. Therefore, higher uncertainty (cid:0) is not only associated with lower mean growth but also contributes to a more disperse and negatively skewed growth distribution. We empirically estimate the distribution of future real growth in industrial production as a function of uncertainty measures using quantile regression methods.1 We document three stylized facts from the estimations. First, higher economic uncertainty is associated with a more dispersed and left-skewed future growth distribution. A one-standarddeviationincreaseinuncertaintystatisticallysigni(cid:133)cantlyincreasestheinterquartilerange of the one-month ahead annualized growth distribution by 2%; and decreases the lower 5th percentile by 5%. This indicates that the marginal e⁄ects of higher uncertainty are 1ThisempiricalmodelisbasedonAdrian,BoyarchenkoandGiannone(2019)(ABG).Insteadofusing uncertainty measure, ABG uses national (cid:133)nancial condition index (NFCI) as the conditional variable to estimate the growth distribution. 1

Mean Growth Growth Volatility Growth Skewness 1.1 0.14 1.5 1 0.12 0.5 1.05 0.1 0 0.5 0.08 1 1 0.06 1.5 0.95 0.04 2 0.6 0.8 1 0.6 0.8 1 0.6 0.8 1 Uncertainty Uncertainty Uncertainty Figure 1: Uncertainty and Higher Moments of Growth. Note: this (cid:133)gure depicts the contemporaneous relationship between JLN macro uncertainty and 36-month rolling window average IP growth (left panel), growth volatility (middle panel), and growth skewness (right panel). The sample spans the period 1971:01 to 2018:12. to signi(cid:133)cantly increase growth downside risk. Second, the response of IP growth to changes in uncertainty is highly asymmetric, the response being much higher when uncertainty rises than when it falls. An increase in uncertainty is associated with a larger decrease in the lower tail of the growth distribution while it has a much smaller impact on its upper tail. These results suggest that higher uncertainty could lead to an abrupt economic decline whereas lower uncertainty does not necessarily rebound the economy from the recession. Third, higher asset volatility magni(cid:133)es the negative impact of uncertainty on growth. We (cid:133)nd that when the asset market is more volatile as measured by higher VIX, an increase in macro uncertainty has a larger negative impact on the lower tail of the expected growth distribution. A one standard deviation increase in VIX increases in the marginal e⁄ect of uncertainty on the lower 5th percentile by 10%. Motivatedbythisempiricalevidence,wepresentandestimateananalyticallytractable model that generate endogenous growth and uncertainty. The equilibrium growth is the result of the adoption of technologies of uncertain quality. A technology(cid:146)s (cid:147)quality(cid:148)refers to how closely the technology(cid:146)s needs match the economy(cid:146)s input endowments. Since technology needs are unpredictable, rapid irreversible adoption of new technologies can lead to high uncertainty and decline in the growth. In equilibrium, uncertainty a⁄ects growth because (cid:133)rms adopt technologies the exact characterofwhichtheydonotknow. Howwellatechnology(cid:133)tsa(cid:133)rm(cid:146)sassetendowments is revealed only after the fact. The mismatch is distributed symmetrically but the cost of 2

that mismatch is quadratic so that maximal losses from technology adoption exceed the maximal gains. The (cid:133)rm(cid:146)s growth rate is therefore negatively skewed as observed in the data. The model provides an analytic characterization of the equilibrium growth distribution. Consistent with our empirical evidence, higher uncertainty about the newly adopted technologies leads to a lower average growth as well as a more dispersed and negatively skewed growth distribution. By assuming that the uncertainty has no long-run impacts on growth, the calibrated model is able to quantitatively match the marginal e⁄ect of uncertainty on several key moments of the growth distribution. As in the data, the model suggests that higher asset price volatility, measured from the estimated option prices of a Lucas (1978) representative security, leads to a higher marginal e⁄ect of uncertainty on the lower tail of the growth distribution. Related Literature Our paper relates closely to two major strands of literature. First, a growing body of empirical studies the real e⁄ect of uncertainty. Carriero, Clark and Marcellino (2018) (cid:133)nds that economic uncertainty has a strong negative e⁄ect on economic outcomes. Similarly, based on breaks in the volatility of macroeconomic variables, Angelini, Bacchiocchi, Caggiano and Fanelli (2019) shows that macro uncertainty has a contractionary e⁄ect on output. Using a shock-restricted SVAR approach, Ludvigson et al. (2019) (cid:133)nds that (cid:133)nancial uncertainty could be a possible source of business cycle (cid:135)uctuation. In spite of the mixed evidence on which type of uncertainty has a contractionary e⁄ect on economic growth, it(cid:146)s evidential that the higher uncertainty about future economy exhibits large impactonexpectedmeangrowth. However, allthesepapersempiricallyinvestigateuncertainty and real variables using structural vector autoregression (SVAR) framework and thus cannot establish the relationship between uncertainty and higher moments of the growth distribution.2 One exception is Hengge (2019). As in our paper, she uses quantile regression analysis and shows that the relationship between macroeconomic uncertainty and future quarterly GDP growth is highly nonlinear and asymmetric. Our empirical evidence on the asymmetric response of expected IP growth to higher uncertainty is consistent with hers despite using monthly industrial production as the measure of economic growth. Using quantileregressionestimates, wefurtherconstructtheGrowth-at-Riskmeasureandstudy 2OtherexamplesincludeCaldara, Fuentes-Albero, GilchristandZakraj(cid:154)ek(2016), Alfaro, Bloomand Lin (2016), and Shin and Zhong (2018). 3

its interaction with asset pricing volatility and capacity utilization both empirically and theoretically. To the best of our knowledge, ours is the (cid:133)rst paper to provide a theoretical framework to study the impact of uncertainty on higher moments of the growth distribution. Second, a large body of theoretical literature proposes that higher economic uncertainty as a cause of the decline in growth. This includes models of the real options e⁄ects of uncertainty (Bernanke (1983), McDonald and Siegel (1986)), models in which uncertainty in(cid:135)uences (cid:133)nancing constraints (Gilchrist, Simand Zakrajsek (2010), Arellano, Bai and Kehoe (2011)), investment (Fajgelbaum, Schaal and Taschereau-Dumouchel (2017)), or precautionary saving (Basu and Bundick (2017), Leduc and Liu (2016), FernÆndez- Villaverde, Pablo Guerr(cid:243)n-Quintana and Uribe (2011)). Our notion of uncertainty in the model is similar to Bloom (2009) and Bloom et al. (2018) that assumes that higher uncertainty originates directly in the process governing technological innovation and adoption. Our model follows the putty-clay tradition of Johansen (1959), assuming irreversible technological commitment. It builds on Jovanovic (2006) by deriving the closed-form expression of equilibrium growth, and studying the relationship between uncertainty and higher moments of the growth distribution.3 The rest of this paper is organized as follows. Section 2 provides some empirical evidence on the asymmetric real e⁄ect of uncertainty. Section 3 presents the model, analyzes the model(cid:146)s empirical implications, and provides an extended model with learning. Section 4 concludes the paper. 2 Empirical Evidence In this section, we describe the data and present the key empirical fact that the relationship between future economic growth and uncertainty is highly nonlinear. To document this feature in the data, we follow Adrian et al. (2019) and investigate the relationship between uncertainty and future growth via forecasting quantile regressions. Compared to the traditional OLS forecasting regressions, quantile regressions describe how the set of conditional variables, including uncertainty measures, a⁄ect di⁄erent quantiles of the future growth. This methodology thus allows the estimated relationship between uncertainty and future growth to di⁄er across quantiles. This extension of a simple linear regression model can capture the potential nonlinear relationship between the shocks in uncertainty and vulnerability of the growth, which is largely neglected in the literature. 3Other related models are Ramey and Ramey (1991), Chalkley and Lee (1998), Veldkamp (2005), Klenow (1998), and Acemoglu and Scott (1997). 4

Moreformally,weregresstheh-month-aheadrealindustrialproductiongrowth(thereafter "IP growth") on a vector of condition variables x , t (cid:1)ip = (cid:14) x +" ; (1) t+h;(cid:11) 0(cid:11);h t t where the conditional variables x include a constant, the lag of the IP growth, and t uncertainty U . t The regression slope can be solved by minimizing the quantile-weighted absolute value of errors: (cid:11)1 (cid:1)ip (cid:14) x ^ (cid:14) = argmin (cid:1)ipt+h;(cid:11)>(cid:14)0(cid:11);h xtj t+h;(cid:11) (cid:0) 0(cid:11) t j ; (2) (cid:11);h +(1 (cid:11))1 (cid:1)ip (cid:14) x t ( (cid:0) (cid:1)ipt+h;(cid:11)<(cid:14)0(cid:11);h xtj t+h;(cid:11) (cid:0) 0(cid:11) t j ) X where h is the forecasting horizon, (cid:11) is the quantile and 1 is the indicator variable. For inference, standard errors are estimated via the bootstrap. 2.1 Data ThemonthlyindustrialproductiondataisobtainedfromFREDEconomicdatabasemaintained by the Federal Reserve Bank of St. Louis. Our main measures of uncertainty U t are constructed following the framework of Jurado et al. (2015) and Ludvigson et al. (2019), which aggregates over a large number of estimated uncertainties constructed from a panel of data. Let yC YC = (yC;:::;yC ) be a variable in category C. Its h-period jt 2 t 1t NCt 0 ahead uncertainty, denoted by C(h), is de(cid:133)ned to be the volatility of the purely un- Ujt forecastable component of the future value of the series, conditional on all information available. Speci(cid:133)cally, Uj C t (h) (cid:17) s E (y j C t+h (cid:0) E[y j C t+hj I t ])2 j I t ; (cid:20) (cid:21) where I denotes the information available. Uncertainty in category C is an aggregate of t individual uncertainty series in the category: NC 1 U Ct (h) (cid:17) plim NC !1 N C Uj C t (h) (cid:17) EC [ Uj C t (h)]: j=1 X If the expectation today of the squared error in forecasting y rises, uncertainty in the jt+h variable increases. In this paper, we focus on macro uncertainty and use a monthly macro dataset, consisting of 134 mostly macroeconomic time series taken from McCracken and 5

)% Median 2 ( e ta 0 R h 2 tw o 4 rG 0 10 20 30 40 50 60 Month )% Interquartile Range 4 ( e ta 2 R h 0 tw o 2 rG 0 10 20 30 40 50 60 Month )% 5th Percentile ( e 0 ta R h 5 tw o 10 rG 0 10 20 30 40 50 60 Month Figure 2: Impact of a one-standard deviation increase in uncertainty measure on the annualized growth rate. Note: this (cid:133)gure report the estimated coe¢ cient of uncertainty over h = 1;2:::;60 months from the baseline quantile regression described in the texts. The bootstrapped 67% con(cid:133)dence intervals are reported in red dashed lines. Ng (2016).4 Following Ludvigson et al. (2019), 1-month ahead macro uncertainty as our baseline measure of uncertainty. Our sample is monthly and spans the period 1971:01 to 2018:12, unless otherwise noted. 2.2 Results By construction, any e⁄ects of changes in x on h-month ahead growth at (cid:11) percentile is t ^ captured by the slope estimate (cid:14) . Figure 2 reports the estimates of marginal coe¢ cient (cid:11);h of uncertainty over h = 1;2:::;60 months. The top panel reports the e⁄ect of uncertainty on changes in expected median growth. We (cid:133)nd that a one-standard-deviation increase in uncertainty decreases the median of the annualized growth by 3% over the next month. The e⁄ect gradually weakens over time with a half-life of 23 months. The middle panel shows that the distribution of the growth becomes more dispersed following an increase in uncertainty. The interquartile range rises at short- and medium- terms. Similarly, the bottom panel shows that when uncertainty increases, the distribution of the growth is more negatively skewed. The 5th percentile of the growth falls by 6% at short horizon, 4Therefore, the category C is set to be the macro category M in Ludvigson et al. (2019). 6

Estimates of Uncertainty Coefficients 2 1 0 1 )% ( e 2 ta R h 3 tw o rG 4 5 percentile 25 percentile 5 Median 75 percentile 6 95 percentile 7 0 10 20 30 40 50 60 Month Figure 3: Asymmetricimpactofuncertaintyongrowth. Note: this(cid:133)gurereportstheestimated uncertianty coe¢ cients for di⁄erent quantiles (cid:11) over horizon h from the baseline quantile regression described in the texts. and 2% over 2 to 4 years. As a result, the impact of the uncertainty on skewness of the distribution is long-lasting. In Figure 3, we plot the estimated uncertainty coe¢ cients (cid:14) for di⁄erent (cid:11) over (cid:11);h horizon h. In most cases, an increase in uncertainty is associated with a larger decrease in the lower tail of the growth distribution while it has much smaller impact on the upper tail. This asymmetric response shows that the impact of uncertainty on growth is highly nonlinear. 2.3 Growth-at-Risk TheGrowth-at-Risk(GaR)inourpaperisde(cid:133)nedastheestimatedlower5thpercentileof the distribution of expected real IP growth. It(cid:146)s worth noting that our measure of GaR is di⁄erent from the one in Adrian, Grinberg, Liang and Malik (2018) because they use the national (cid:133)nancial condition index (NFCI) from the Chicago Fed in conditional variables x whereas we include the uncertainty measure.5 Figure 4 shows the time series of onet month ahead GaR estimated using uncertainty measure. The red line in the (cid:133)gure shows 5In addition, Adrian et al. (2018) uses quarterly GDP growth whereas we use monthly industrial production as the measure of economic growth. 7

the he GaR estimated using NFCI for comparisons. First, it shows that both measures of GaR are counter-cyclical and with a less than 5% probability, a one-standard deviation increase in uncertainty is associated with as large as 3% decline in IP Growth within a month during recessions. Second, GaR estimated with uncertainty is more volatile and exhibit larger spikes during recessions. For the rest of the paper, we refer to GaR as the one that is estimated using uncertainty. One month ahead Growth at Risk (%) 1 Macro Uncertainty 0.5 NFCI 0 0.5 1 1.5 2 2.5 3 3.5 4 1970 1980 1990 2000 2010 2020 Figure 4: Growth-at-Risk. Note: this(cid:133)gurereportstheestimatedGrowth-at-RiskusingNFCI(red line) and JLN Macro uncertainty (black line). The baseline quantile regression is described in the texts. The sample spans the period 1971:01 to 2018:12. Our baseline measure of uncertainty is based on macro variables but some papers in the literature, such as Ludvigson et al. (2019), found evidence that volatility related to theassetreturnsalsoleadtopersistentdeclineIPgrowth. Thisevidencesuggeststhatthe uncertainty should be associated with a larger decline in growth when the asset prices are more volatile. To directly test this argument, Figure 4 plots the estimated GaR against the standardized implied volatility indices (VIX), which is a popular measure of the stock market(cid:146)s expectation of volatility implied by S&P 500 index options.6 It shows that when asset market is more volatile, an increase in macro uncertainty has a larger negative impactonIPgrowthandthusleadstoalowerGaR.Onaverage, atwo-standard-deviation 6In the paper, we chose to use the VXO implied volatility index that is available since 1962 instead of VIX because the latter is only available after 1990 and the two measures are 99% correlated. 8

increase in VIX is associated with a 0.5% decline in GaR. VIX and Growth at Risk 0 0.5 ) % ( k 1 s iR t a 1.5 h t w o r G 2 2.5 3 2 1 0 1 2 3 4 5 6 7 VIX (Standarized) Figure 5: Growth-at-Risk and VIX. Note: this (cid:133)gure shows the contempraneous relationship between standardized VIX and estimated Growth-at-Risk. The Growth-at-Risk is estimated conditional onJLNmacrouncertainty. TheredlinereportstheOLSEstimate. Thesamplespanstheperiod1971:01 to 2018:12. To sum up, we document some empirical evidence on the real e⁄ect of uncertainty. First, we (cid:133)nd that an increase in uncertainty leads to lower median and yet more volatile IP growth, and a more negative GaR. Second, the response of IP growth to an increase in uncertainty is highly nonlinear, especially at the short term. Third, higher asset volatility magni(cid:133)es the negative e⁄ect of uncertainty on growth. In the next section, we set up a tractable model that is able to capture these key features in the data. 3 Model The model features a single agent (cid:147)Crusoe(cid:148), his preferences over consumption sequences (c ) are t 1 E (cid:12)tlnc : 0 t ( ) 0 X Production (cid:150)For simplicity of the exposition, we assume there is a single production good y and there is no physical capital. The output depends on the level of technology A 9

as well as Crusoe(cid:146)s skill mix h. The potential output, yp, is de(cid:133)ned as (cid:21) yp = exp A (s h) 2 : A 0 (cid:0) 2 (cid:0) (cid:26) (cid:27) where s is the skill-mix ideal for technology A. By construction, (cid:21) (s h)2 captures A 2 A (cid:0) 0 the foregone output due to the "skill gap" between Crusoe(cid:146)s current skill and the ideal skill for technology A. Adoption of Technology (cid:150)Crusoe can raise his technology by any amount, x 0, (cid:21) so that starting today at A; tomorrow(cid:146)s technology is A = A+x: 0 Crusoe must use technology A for at least one period. We assume that adopting a new 0 technology is free but adoption of the new technology A makes unpredictable demands 0 on the skill mix, s = s +x"; (3) A A 0 where " F (") is time speci(cid:133)c and i.i.d., having mean zero and variance (cid:27)2. Once " is (cid:24) drawn, s becomes an invariant skill requirement for technology A. Crusoe chooses A A 0 0 0 before seeing ", and he cannot return to technologies that he used in the past. Adjustment of h.(cid:151)Crusoestartstheperiodwithh. Beforeproducing,hecanchange it to h h+(cid:1) at a adjustment cost of 0 (cid:17) (cid:18) C(yp;(cid:1)) 1 exp (cid:1)2 yp: (cid:17) (cid:0) (cid:0)2 (cid:20) (cid:26) (cid:27)(cid:21) Finally, Crusoe(cid:146)s net output is (cid:21) (cid:18) yp C(yp;(cid:1)) y(u;(cid:1);A) = exp A (u (cid:1))2 (cid:1)2 ; (cid:0) (cid:17) (cid:0) 2 (cid:0) (cid:0) 2 (cid:26) (cid:27) where u = s h; (4) A (cid:0) is the gap between ideal and actual skill (hereafter the (cid:147)skill gap(cid:148)). 3.1 Optimal Skill Investment The state consists of the technology level and his skill gap (u;A), and his decisions are (x;(cid:1)). He has no assets other than h and A, and he simply consumes his output. His Bellman equation is V (u;A) = max lny(u;(cid:1);A)+(cid:12) V (u+x" (cid:1);A+x)dF (") : (5) x;(cid:1) (cid:0) (cid:26) Z (cid:27) The solution to (5), derived in Appendix 1, can be summarized as follows: 10

Proposition 1 (Jovanovic 2006) The policy functions are 1 x = ; (6) (cid:18)(cid:27)2(1 (cid:12))(1 (cid:11)) (cid:0) (cid:0) and (cid:1) = (1 (cid:11))u; (7) (cid:0) where 1 (cid:21) (cid:21) 2 (cid:11) = 1+(cid:12) + 1+(cid:12) + 4(cid:12) ; (8) 2(cid:12) 8 (cid:18) (cid:0)s (cid:18) (cid:0) 9 (cid:18) (cid:19) < = is the fraction of the gap that Crusoe leaves open. The solution for V is : ; A 1 V (u;A) = (cid:18)(1 (cid:11))u2 +(cid:7); (9) 1 (cid:12) (cid:0) 2 (cid:0) (cid:0) where (cid:12) x 1 (cid:7) = x2(cid:27)2 ; and = (cid:21)(cid:11)2 +(cid:18)(1 (cid:11))2 : (10) 1 (cid:12) 1 (cid:12) (cid:0) 1 (cid:12)(cid:11)2 2 (cid:0) (cid:0) (cid:18) (cid:0) (cid:20) (cid:0) (cid:21) (cid:19) (cid:0) (cid:1) Proposition 1 states that Crusoe(cid:146)s adjustment of skills (cid:1) is proportional to his current skill gap u. As a result, using equation (3) and (4), the skill gap u follows a AR(1) process as follows, u = (cid:11)u +x" : (11) t+1 t t+1 Uncertainty (cid:150)Our notion of uncertainty is captured by the variance of the skill gap u. Because (cid:11) is between zero and one, u is stationary and its variance is t x2(cid:27)2 (cid:28) Var(u) = = (1 (cid:12))2 1 (cid:11)2 (cid:18)2(1 (cid:11))2 (cid:0) 1 (cid:27)2; (cid:17) 1 (cid:11)2 (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:1) (cid:1) wherethesecondequalitycomesfromequation(6). Conditionalonthecurrentskillgapu, an increase in Crusoe(cid:146)s skill investment, through lower (cid:11) or (cid:18), leads to lower uncertainty. In addition, noisier " (higher (cid:27)2) leads to higher uncertainty. Free-riding incentives and decentralization of the optimum (cid:151)If all agents start in the same state and then take the same action, the equilibrium is symmetric and it has only aggregate risk. Section 4 of Jovanovic (2006) analyzes the incentives of agents to deviate from this situation by considering the option to (cid:147)wait and see(cid:148)how other (cid:133)rms fare with the newly adopted technology. The decentralized version assumes that there are two markets: A market for output, and a market for (cid:133)rm(cid:146)s shares (cid:150)the only assets available to households. No markets exist for either A which, since it was free for Crusoe to augment, naturally would also be freely copied in a decentralized setting, and yet (cid:133)rms 11

still reject the option to wait and see.7 On the other hand, a (cid:133)rm cannot buy h from other (cid:133)rms; one interpretation of h is that of organization capital in the sense of Prescott and Visscher (1980). 3.2 Growth Distribution Suppose that C(yp;(cid:1)) consists entirely of foregone output. The log of measured output then is, (cid:21) (cid:18) lny = A (cid:11)2u2 (1 (cid:11))2u2 (12) t (cid:0) 2 t (cid:0) 2 (cid:0) t = A +xt u2, 0 (cid:0) t Because u is stationary, lny is trend-stationary, the trend and the long-run rate of output t growth is x. The log growth can be expressed as, g lny lny = x u2 u2 : (13) t+1 (cid:17) t+1 (cid:0) t (cid:0) t+1 (cid:0) t (cid:0) (cid:1) Equation (13) shows that the growth is driven not just by the adoption of newtechnology, x; as studied in the traditional macro literature. It is also by the changes in Crusoe(cid:146)s skill gap u2 u2. On one hand, a decrease in the skill gap stimulates the growth. On the t+1 (cid:0) t other hand, adjusting h towards its technologically ideal value is costly and will therefore take time, but output will fall sharply whenever an unlucky draw of s occurs. So we A should expect the growth distribution is negative skewed especially when (cid:18) is large. This leads to the following proposition. Proposition 2 Condition on initial output and technology level y and A , if " follows a 0 0 t normal distribution N (0;(cid:27)2), the distribution of growth g is left skewed and is expressed t as g = x C + 1 (cid:11)2 (cid:28)(cid:24)2 +((cid:11)A B )(cid:11)tu (cid:24) ; t (cid:0) t (cid:0) t t (cid:0) t 0 t 7One could add features that w(cid:2)ould im(cid:0)ply a c(cid:1)oexistence of heterogeneous t(cid:3)echnologies in a puttyclay framework. These include a labor market (Johansen (1959)) and (cid:133)rm-speci(cid:133)c shocks to the cost of investment (Campbell (1998) and Gilchrist and Williams (2000)). These features would allow the model to match facts about technology-di⁄usion lags. 12

where (cid:24) N (0;1) follows standard normal distribution and t (cid:24) 1 (cid:11)2t A = 4(1 (cid:11)2)(cid:28) (cid:0) +1 t s (cid:0) 1 (cid:11)2 (cid:18) (cid:0) (cid:19) 1 (cid:11)2t B = 4(1 (cid:11)2)(cid:28) (cid:0) t s (cid:0) 1 (cid:11)2 (cid:18) (cid:0) (cid:19) C = (cid:11)tu 2 (cid:11)t 1u 2 t 0 (cid:0) 0 (cid:0) 1 u = (cid:0) ( (cid:1) lny (cid:0) A )2: (cid:1) 0 0 0 (cid:0) r Because (cid:24) follows a normal distribution, (cid:24)2 follows the chi-squared distribution with t t the degree of freedom of one. Since > 0, the distribution of (cid:1)lny is negatively skewed. t The following corollary characterizes the (cid:133)rst two moments of the growth distribution, such mean, median, variance and interquartile range (IQR). Corollary 1 In equilibrium, the distribution of growth g satis(cid:133)es t+1 E(g ) = x C 1 (cid:11)2 (cid:28) t+1 t+1 (cid:0) (cid:0) (cid:0) Median(g t+1 ) = x C t+1 0:4(cid:0)7 1 (cid:1) (cid:11)2 (cid:28) (cid:0) (cid:0) (cid:0) V(g t+1 ) = 2 2 1 (cid:11)2 2 (cid:28)2 + (cid:0) 2 ((cid:11)A(cid:1)t+1 B t+1 )(cid:11)tu 0 2 (cid:0) (cid:0) IQR(g t+1 ) = 1(cid:0):22 1 (cid:1)(cid:11)2 (cid:28) +1:(cid:0)34((cid:11)A t B t )(cid:11)tu 0 :(cid:1) (cid:0) (cid:0) (cid:0) (cid:0) (cid:1) (cid:1) Mean and median of the growth distribution are decreasing in uncertainty (cid:28) whereas the variance and IQR increases in uncertainty, @E(g t+1 ) @Median(g t+1 ) @V(g t+1 ) @IQR(g t+1 ) < 0, < 0; > 0, > 0: @(cid:28) @(cid:28) @(cid:28) @(cid:28) As uncertainty increases, the average and median growth decreases while the growth distributionbecomesmoredispersed. Inthelongrun, we(cid:133)ndthatthegrowthdistribution doesn(cid:146)t depend on the initial values of u and measures of centrality and dispersion of 0 the growth distribution are bounded as long as u is stationary. The following corollary t summarizes this (cid:133)nding. Corollary 2 When t , the (long run) distribution of growth g doesn(cid:146)t depend on t ! 1 the initial condition u and satis(cid:133)es 0 g = x x2(cid:27)2(cid:24)2 (14) t (cid:0) t 13

The (cid:133)rst two moments of the growth distribution are bounded in the limit and satisfy the following expressions if and only if (cid:11) < 1; lim E(g) = x 1 (cid:11)2 (cid:28) t (cid:0) (cid:0) !1 lim V(g) = 2 2 1(cid:0) (cid:11)2 2(cid:1)(cid:28)2 t (cid:0) !1 lim Median(g) = x (cid:0)0:47 1(cid:1) (cid:11)2 (cid:28) t (cid:0) (cid:0) !1 lim IQR(g) = 1:22 1 (cid:0)(cid:11)2 (cid:28): (cid:1) t (cid:0) !1 (cid:0) (cid:1) Growth-at-Risk (GaR)(cid:150)FollowingAdrianetal.(2018), theGrowth-at-Risk(GaR) at time t is de(cid:133)ned as the lower 5th percentile of the growth distribution g . Using the t+1 fact that 95th percentile of the standard normal is 1:65 and the 95th percentile of chisquare is 3:84, GaR (cid:31) in our model can be expressed as t (cid:31) = x C 3:84 1 (cid:11)2 (cid:28) +1:65 ((cid:11)A B )(cid:11)tu ; (15) t t+1 t+1 t+1 0 (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:1) In equilibrium, GaR decreases in uncertainty ((cid:28)); @(cid:31) t < 0: @(cid:28) Consistent with the empirical (cid:133)nding, larger uncertainty is associated with higher downside risk. 3.3 Impact of Uncertainty on Growth We now examine the impact of uncertainty on growth. Starting with (cid:27) = 1; we raise the (cid:27) to 1:05 and 1:15 so that (cid:28), the variance of u, increases by 15% and 30%, respectively. A 15% increase in uncertainty corresponds to a one-standard-deviation increase in the empirical measure of uncertainty. In order to capture the transitory impact of uncertainty as in the empirical section, we further restrict parameters so that the uncertainty shock does not have a permanent e⁄ect on the distribution of the growth rate. More speci(cid:133)cally, we adjust the parameter pair ((cid:21);(cid:18)) so that the long-run mean and variance of the growth rate do not change with their long-run values remaining at lim E(g ) = x x2(cid:27)2 = 1:5%, and t+1 t (cid:0) !1 lim V (g ) = 2 2x4(cid:27)4 = 1%: t+1 t !1 14

We simulate the model for 60 periods and compute the changes in growth relative to the case that (cid:27)2 is (cid:133)xed at 1. Figure 6 depicts the response of median growth in the top panel , interquartile range in the middle panel and 5th percentile (GaR) in the bottom panel to a unexpected increase in uncertainty at time 0. The blue line depicts the case with small uncertainty((cid:27) = 1:05) while the redline reports results withlargeuncertainty((cid:27) = 1:15). The dotted black line depicts the data counterpart reported in Figure 2. Similar to the empirical evidence, an increase in uncertainty is associated with a lower median growth, higher interquartile range, and lower Growth-at-Risk. Quantitatively, our model suggests that one standard deviation increase in uncertainty immediately results in 2% decline in median growth, 3%increase in the growth dispersion and a 6%decline in GaR. Compared to the data, our model (cid:133)ts the impact response well but underpredicts (overpredicts) the median and GaR (interquartile range) in the medium term. This suggests the e⁄ect of uncertainty is more persistent in the data. Median )% 0 ( e ta R 2 h tw 4 o r G 0 10 20 30 40 50 60 Interquartile Range )% ( 4 Small Uncertainty (15%) e ta Large Uncertainty (30%) R2 Data h tw0 o r G 0 10 20 30 40 50 60 5th Percentile )% 0 ( e ta R 5 h tw o 10 r G 0 10 20 30 40 50 60 Month Figure 6: Uncertainty and Growth. Note: this (cid:133)gure depicts the response of median growth in the top panel , interquartile range in the middle panel and 5th percentile (GaR) in the bottom panel to an unexpected 15% (blue line) and 30% increase (red line) in uncertainty at time 0. The dashed black line is the empirical estimates from the baseline quantile regression described in the texts. Figure7depictstheresponseofvariousquantilesoftheexpectedgrowthdistributionto 15

Response to 15% increase in Uncertainty (Model) Response to 15% increase in Uncertainty (Data) 2 2 1 1 0 0 )% 1 )% 1 ( ( e ta e ta 2 R 2 R h tw h tw 3 o r 3 o r G 5 percentile G 4 5 percentile 4 25 percentile 25 percentile 5 Median Median 5 75 percentile 6 75 percentile 95 percentile 95 percentile 6 7 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Month Month Figure 7: Asymmetric impact of uncertainty on growth. Note: the left panel depicts the response of various quantiles of the expected growth distribution to a 15% increase in uncertainty. The right panel reports the the empirical estimates from the baseline quantile regression described in the texts. a15%increaseinuncertainty((cid:27) = 1:05). Consistent withtheempirical evidencereported in Figure 3, the response of the expected growth is highly nonlinear: an increase in uncertainty is associated with a larger decrease in the lower tail of the growth distribution whereas it has much smaller impact on the upper tail. Quantitatively, a 15% increase in uncertainty results in 4% decrease in 25th percentile of the expected growth distribution but only contributes to a 1% decline of the 75th percentile. 3.4 Growth-at-Risk, the value of options and VIX For the price of a representative security p(u;A) we have the Lucas (1978) equation c(u;A) p(u;A) = (cid:12) [c(u;A+x)+p(u;A+x)]d(cid:8); (16) 0 0 c(u;A+x) Z 0 because U 0 (c(u 0 ;A+x)) = c(u;A) . Although preferences being log, prices will (cid:135)uctuate U (c(u;A)) c(u;A+x) 0 0 because consumption growth is autocorrelated. If low consumption today means high consumption growth, a disaster is accompanied by very low asset prices. A put option8 is 8From Investopedia: A put option is an option contract giving the owner the right, but not the obligation, to sell a speci(cid:133)ed amount of an underlying security at a speci(cid:133)ed price within a speci(cid:133)ed time frame. . 16

more valuable in such states, i.e., more likely to be (cid:147)in the money.(cid:148)The opposite for call options. Therefore today(cid:146)s prices of those two derivative assets should be a good signal of Growth-at-Risk. Taking today(cid:146)s price as the strike price for both assets, pput = (p(u;A) p(u;A+x))d(cid:8) (17) 0 (cid:0) Zp(u 0 ;A+x) (cid:20) p(u;A) pcall = (p(u;A+x) p(u;A))d(cid:8): (18) 0 (cid:0) Zp(u 0 ;A+x) (cid:21) p(u;A) The VIX index used in the empirical exercise contains put and call options on the individual (cid:133)rms in the S&P 500 index, and its price would re(cid:135)ect idiosyncratic as well as aggregate risk. The natural de(cid:133)nition in our model is the sum of the prices of a call option and a put option with a strike price tomorrow equal to today(cid:146)s price of the security p(u;A), VIX(u;A) pput +pcall: (19) (cid:17) We do not have idiosyncratic risk in the model so the cross section of asset prices is degenerate. We also are not matching the VIX with the volatility of the price of capital. Instead, we are using the pricing kernel to price the put and call options.9 As before, we increase (cid:27) and adjust the parameter pair ((cid:21);(cid:18)) so that the long-run mean and variance of the growth rate do not change with their long-run values. We then simulate the model and calculate the price the security from equation (16), and price of options and VIX according to equations (17) to (19). The calculated VIX is further standardized to have zero mean and unit variance as in the empirical exercise. Three series are plotted in Figure 8: the red line portrays data regression slope of GaR in the data on the VIX, while yellow and purple lines depict model-generated values under 15% and 30% increase in uncertainty (cid:28), respectively. It shows that an increase in the VIX decreases the GaR in the model for both values of (cid:27). This is consistent with the feature in the data that higher asset price volatility magni(cid:133)es the negative impact of uncertainty on IP growth. In addition, Figure 8 also shows that VIX has larger negative impact on GaR when there is higher uncertainty captured by (cid:27). Compared to a naive empirical linear regression, our model produces a better (cid:133)t for the relationship between VIX and GaR: it has 38% and 14% smaller root-mean squared error for (cid:27) = 1:05 and 9It(cid:146)s worth noting that if the goal is to match stock market values using the price of (cid:133)rm capital, (cid:133)rm leverage should be taken into account (e.g. Bianchi, Ilut and Schneider (2017)). 17

0 0.5 1 ) % ( k 1.5 s iR ta 2 h t w o r 2.5 G 3 Data Empirical Regression 3.5 Model with = 1.05 Model with = 1.15 4 2 1 0 1 2 3 4 5 6 7 VIX (Standarized) Figure 8: Asset Volatility and Growth-at-Risk. Note: The yellow (purple) reports modelgenerated contempraneous relationship between standardized asset volatility and Growth-at-Risk under 15%(30%)increaseinuncertainty. Thebluedotsdepicttheempiricalrelationshipbetweenstandardized VIX and Growth-at-Risk. The red line reports the OLS Estimate. (cid:27) = 1:15, respectively.10 3.5 Capacity Utilization The model has no unemployed resources or spare capacity, but we may assume that the skill gap u is an index of the fraction of capital (cid:150)human or physical (cid:150)that does not meet t the needs of the date-t technology. That interpretation of u would match Ljungqvist and t Sargent(1988)explanationforhighEuropeanunemploymentinthelasttwodecadesofthe 20th century in which higher unemployment re(cid:135)ected restructuring from manufacturing to services, adoption of new information technologies, and globalization. And since Klein and Su (1979) a number of studies show a positive correlation between unemployment and capital utilization measures. 10The root-mean-squared error (RMSE) is de(cid:133)ned as T 1 RMSE = (y^ y )2 vT i (cid:0) i u i=1 u X t where the "hat" refers to the estimated value of GaR at time t. 18

From equation (12), the detrended output in the model depends on the skill gap u . While it(cid:146)s empirically challenging to directly estimate the skill gap in the data, the t capacity utilization rate (cid:150)the ratio of actual output to the potential output (cid:150)can be used to test the relationship between u and the growth. The monthly data on U.S. capacity utilizationisavailableattheFREDeconomicdatamaintainedbytheSt. LouisFed(series id TCU). As before, we simulate the model using calibrated parameters, and calculate skill gap u according to equation (11). We de(cid:133)ne the capacity utilization as 1= u so t t j j that an increase in the skill gap leads to a fall in capacity utilization. The calculated capacity utilization is further standardized to have zero mean and unit variance. 0 0.5 1 ) % ( 1.5 k s iR ta 2 h tw o r G 2.5 3 Data 3.5 Model with = 1.05 Model with = 1.15 4 4 3 2 1 0 1 2 3 Capacity Utilization(Standardized) Figure 9: Capacity Utilization and Growth-at-Risk. Note: The red (yellow) reports modelgenerated contempraneous relationship between standardized capacity utilization and Growth-at-Risk under 15% (30%) increase in uncertainty. The blue dots depict the empirical counterpart. Figure 9 plots model implied Growth-at-Risk (cid:31) on the standardized capacity utilizat tion for (cid:27) = 1:05 and 1:15, respectively. It shows that, consistent with data, an decrease in the skill gap u or an increase in capacity utilization increases the GaR. In addition, t when the capacity utilization is high, an increase in uncertainty ((cid:27)) has smaller negative impact on growth: a 10% increase (cid:27) decreases GaR by 0.1% when capacity utilization is one standard deviation above its unconditional mean whereas decreases GaR by 0.9% when it is negative one standard deviation below its unconditional mean. 19

3.6 Long Run Growth Distribution Based on equation (14), growth distribution follows a negative chi-squared distribution in the long run. The top panel of Figure 10 depicts the simulated long run distribution for di⁄erent values of (cid:27). We again use calibrated ((cid:21);(cid:18)) such that the mean and variance of the long run distribution remain the same. It shows higher uncertainty (higher (cid:27)) leads to a more negatively skewed distribution (portrayed by red bars) and lower 5th percentile (red dashed line). The chi-squared distribution is well known as a light-tailed distribution whereas in the traditional growth literature (e.g. Barro and Jin (2011)), the heavy-tailed distribution, such as the Pareto Distribution, has been commonly used to calculate the tail risk. To address this issue, we assume that in the long horizon, the growth distribution follows, g = x x2(cid:27)2z : t t (cid:0) The variable z [0; ) follows a Type-II Pareto Distribution with associated CDF, t 2 1 F (z ) = 1 (1+x) (cid:11); t (cid:0) (cid:0) where (cid:11) is the shape parameter that governs the tail thickness. We randomly draw 10,000 z from F (z ) and numerically calculate (cid:11) by targeting t t the lower 5th percentile to be the corresponding value in the calibrated Chi-squared distribution with (cid:27)2 = 1. The bottom panel of Figure 10 depicts the simulated growth distribution using Pareto distribution with (cid:11) = 2:83 (in red bars) against the baseline growth distribution with chi-squared distribution. It shows that Pareto distribution has a thicker tail than the Chi-squared distribution. Therefore, in order to generate the same downside risk, measured by the lower 5th percentile, the growth distribution with Pareto has much smaller mean growth (0:78%) than the chi-squared distribution (1:50%). 3.7 Technological commitment It(cid:146)sworthnotingthattheassumptionofirreversibletechnologicalcommitmentisessential to generate the negatively skewed growth distribution in equilibrium: output could fall because of commitment to technology before an unfavorable shock " is realized. If a (cid:133)rm could revert quickly and costlessly to technologies it used before, it would always use the best technology up to date and output never declines. For example, Jovanovic and Rob (1990) assume costless recall of past technologies. In contrast to our empirical evidence, insteadof havingalefttail, thedistributionof growthratesintheirmodel exhibitsaspike at zero and a right tail. Therefore we need at least partial technological commitment. 20

Figure 10: Simulated Long-run Growth Distribution. Note: the upper panel shows the simulated long-run growth distribution for small uncertainty (blue bars) and large uncertainty (red bars). The lower panel shows the simulated long-run growth distribution with Pareto distribution (red bars) and Chi-squared distribution (blue bars). 3.8 Extension with Learning In the baseline model, we assume that the agent cannot directly learn the ideal skill. We relax this assumption in this section by adding the learning into the model and show that the qualitative result remains the same. As in the baseline model, the ideal skill s is unobservable and the potential output follows, (cid:21) yp = exp A (s h)2 ; (cid:0) 2 (cid:0) (cid:26) (cid:27) where for simplicity we assume that the ideal skill s is the sum of two components s = (cid:23) +" : t (cid:23) R is a parameter and " is i.i.d. with zero mean and variance of (cid:27) . We further t " 2 assume agent(cid:146)s prior over " is di⁄use. t 21

Each period, the agent can purchase n signals of parameter " with a cost (in foregone t output), (cid:18) C(yp;n) 1 exp n2 yp: (cid:17) (cid:0) (cid:0)2 (cid:20) (cid:26) (cid:27)(cid:21) The net output therefore is (cid:21) (cid:18) yp C(yp;n) = exp A u2 n2 ; (cid:0) (cid:0) 2 (cid:0) 2 (cid:26) (cid:27) where u = s h is the skill gap, and where n is the number of independent signals on (cid:0) " ; call these signals, (cid:24) ;:::;(cid:24) . Each (cid:24) is an independent signal on next period(cid:146)s " t+1 1 n;t i as follows: (cid:0) (cid:1) (cid:24) = " +(cid:17) ; i t+1 i where(cid:17) N 0;(cid:27)2 arei.i.d. randomvariables. Theagentformsthemeanofthesample i (cid:24) (cid:17) of n signals: (cid:0) (cid:1) n n 1 1 (cid:22) (cid:24) = (cid:24) = " + (cid:17) : (20) n n i t+1 n i i=1 i=1 X X Since each period " is independent of its past draws, (cid:22) (cid:24) N ";(cid:27)2=n is a su¢ cient n (cid:24) (cid:17) (cid:22) statistic for forecasting ": And since the prior over " is di⁄use, conditional on (cid:24) , " (cid:0) (cid:1) n (cid:24) N (cid:22) (cid:24) ;(cid:27)2=n . More formally, n (cid:17) (cid:0) (cid:1) (cid:22) " (cid:24) lim Pr " (cid:22) (cid:24) = (cid:8) (cid:0) n ; (21) (cid:27)" !1 j n (cid:18) (cid:27)2 (cid:17) =n (cid:19) (cid:0) (cid:1) where (cid:8) is the standard normal integral. Denote (cid:28) the beginning-of-period prior variance over (cid:23); and (cid:28) the value next period. 0 Then we have, 1 n 1 (cid:0) (cid:28) = + b((cid:28);n); (22) 0 (cid:28) (cid:27)2 (cid:17) (cid:18) (cid:17)(cid:19) which shows the persisting bene(cid:133)ts of n via the reduction in (cid:28). As in Fajgelbaum et al. (2017), thenotionof uncertaintyiscapturedbythevarianceof beliefs(cid:28). Itslawof motion is speci(cid:133)ed in equation (22). In equilibrium, the agents set h optimally to be the posterior mean and the next period(cid:146)s skill gap follows a normal distribution (cid:27)2 (cid:17) u N 0;(cid:28) + ; 0 (cid:24) n (cid:18) (cid:19) 22

or alternatively, (cid:27)2 1=2 (cid:17) u = (cid:28) + (cid:16); 0 n (cid:18) (cid:19) where (cid:16) follows a standardized normal distribution. Log output can be expressed as (cid:21) (cid:18) lny = A u2 n2: (23) (cid:0) 2 (cid:0) 2 It(cid:146)s straightforward to verify that the state variable are (u;(cid:28)), which follow the law of motion, (cid:27)2 1=2 (cid:17) u = (cid:28) + (cid:16) 0 n (cid:18) (cid:19) 1 n 1 (cid:0) (cid:28) = + : 0 (cid:28) (cid:27)2 (cid:18) (cid:17)(cid:19) The value function can be expressed as (cid:21) (cid:18) 1 n 1 (cid:27)2 1=2 V ((cid:28);u) = maxA u2 n2 +(cid:12) V + (cid:0) ; (cid:28) + (cid:17) (cid:16) dF ((cid:16)): n (cid:0) 2 (cid:0) 2 Z (cid:18) (cid:28) (cid:27)2 (cid:17)(cid:19) (cid:18) n (cid:19) ! s:t: V ((cid:28);u) 0 (cid:21) Using equation (23), growth can be expressed as, (cid:21) (cid:1)lny = u2 u2 t+1 2 t (cid:0) t+1 (cid:21) (cid:0) (cid:1) (cid:27)2 = u2 (cid:28) + (cid:17) (cid:16)2 : 2 t (cid:0) n (cid:18) (cid:18) (cid:19) (cid:19) First, we observe that the economy growth is driven by uncertainty though changes in the skill gap. Reducing the skill cap increases the growth and vice versa. Second, higher uncertainty leads to lower growth, @(cid:1)lny (cid:21) t+1 = (cid:16)2 < 0: @(cid:28) (cid:0)2 Third, conditional on current skill gap u , the distribution of the growth is left skewed t with the skewness, (cid:27)2 @skewness((cid:1)lny ) (cid:17) t+1 skewness((cid:1)lny ) = (cid:21) (cid:28) + = (cid:21): t+1 (cid:0) n ) @(cid:28) (cid:0) (cid:18) (cid:19) Similartothebaselinemodelwithoutlearning, higheruncertaintyleadstohighernegative skewness and lower Growth-at-Risk. In the appendix, we provide an analytical solution to the distribution of growth in the equilibrium when the parameter (cid:23) is known. 23

4 Conclusion In this paper, we (cid:133)rst empirically documented several stylized facts on the real e⁄ect of uncertainty. First, we showed that higher economic uncertainty is closely associated with a more dispersed and left-skewed growth distribution. Second, we found that the response of IP growth to an increase in uncertainty is highly nonlinear and asymmetric. Third, we presented evidence that higher asset volatility magni(cid:133)es the negative impact of uncertainty on growth. We presented and estimated a model in which rapid adoption of new technology may lead to higher economic uncertainty which causes measured productivity to decline. The model is able to match several key features in the data. The equilibrium growth distribution is negatively skewed and higher uncertainty leads to higher downside risk. The model is also able to generate a negative nonlinear relation between asset price volatility and Growth-at-Risk in equilibrium as observed in the data. 24

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Appendix 4.1 Proof of Proposition 1 The (cid:133)rst-order condition for the optimality of (cid:1) is (cid:21)(u (cid:1)) (cid:18)(cid:1) (cid:12) V dF = 0; (24) 1 (cid:0) (cid:0) (cid:0) Z and the (cid:133)rst-order condition for the optimality of x is ("V +V )dF = 0: (25) 1 2 Z Solving for (cid:1).(cid:151)The envelope theorem gives V = (cid:21)(u (cid:1))+(cid:12) V dF = (cid:18)(cid:1); (26) 1 1 (cid:0) (cid:0) (cid:0) Z where the second equality uses (24). Substituting into (24) we have (cid:21)(u (cid:1)) = (cid:18)(cid:1) (cid:12)(cid:18) (cid:1)dF ("): (27) 0 (cid:0) (cid:0) Z We seek a solution of the form (7) where (cid:11) is a constant to be solved for. If (7) holds, (27) reads (cid:21)(cid:11)u = (cid:18)(1 (cid:11))u (cid:12)(cid:18) (1 (cid:11))((cid:11)u+x")dF (") (cid:0) (cid:0) (cid:0) Z = (cid:18)(1 (cid:11))u (cid:12)(cid:18)(1 (cid:11))(cid:11)u; (cid:0) (cid:0) (cid:0) which, after cancellation of u leaves a quadratic in (cid:11), namely (cid:18)(1 (cid:11)) (cid:12)(cid:18)(1 (cid:11))(cid:11) (cid:0) (cid:0) (cid:0) (cid:0) (cid:21)(cid:11) = 0, or (cid:21) (cid:12)(cid:11)2 1+(cid:12) + (cid:11)+1 = 0: (28) (cid:0) (cid:18) (cid:18) (cid:19) This implicit function has the solution for (cid:11) given in (8). Solving for x.(cid:151)The envelope theorem also gives 1 V = 1+(cid:12) V dF = : (29) 2 2 1 (cid:12) Z (cid:0) The second equality follows because the right-hand side of (29) is a contraction map with at most one solution for V . Substituting from(7) into (26) and fromthere (in an updated 2 form) into (25) gives 1 0 = "(cid:18)(cid:1)dF + 0 (cid:0) 1 (cid:12) Z (cid:0) 1 = "(cid:18)(1 (cid:11))((cid:11)u+x")dF + ; (30) (cid:0) (cid:0) 1 (cid:12) Z (cid:0) 28

because (cid:1) = (1 (cid:11))u = (1 (cid:11))((cid:11)u+x"). But E("u) = 0, which leads to (6). 0 0 (cid:0) (cid:0) We must show that this function solves (5). Let us proceed with the method of undetermined coe¢ cients. Since V = aA bu2 +c, (cid:0) aA bu2 +c = A u2 +(cid:12) a(A+x) b(u+x" (cid:1))2 +c dF (") (cid:0) (cid:0) (cid:0) (cid:0) Z (cid:0) (cid:1) = A u2 +(cid:12) a(A+x) b((cid:11)u+x")2 +c dF (") (cid:0) (cid:0) Z (cid:0) (cid:1) = A u2 +(cid:12)(a(A+x)+c) (cid:12)b ((cid:11)u+x")2dF (") (cid:0) (cid:0) Z = A u2 +(cid:12)(a(A+x)+c) (cid:12)b(cid:11)2u2 (cid:12)bx2(cid:27)2: (cid:0) (cid:0) (cid:0) Equating coe¢ cients: a = 1+a(cid:12), b = +b(cid:12)(cid:11)2, and c = (cid:12)(ax+c bx2(cid:27)2), so that (cid:0) 1 (cid:12) a = , b = , and c = ax bx2(cid:27)2 : 1 (cid:12) 1 (cid:12)(cid:11)2 1 (cid:12) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:1) This leads to (cid:12) x x2(cid:27)2 V (u;A) = A u2 + 1 (cid:0) (cid:12) (cid:0) 1 (cid:0) (cid:12)(cid:11)2 ; 1 (cid:12) (cid:0) 1 (cid:12)(cid:11)2 (cid:16) 1h (cid:12) i (cid:17) (cid:0) (cid:0) (cid:0) where x = 1 . Then V (u;A) = 1=(1 (cid:12)) which is consistent with (29). It re- (cid:18)(cid:27)2(1 (cid:12))(1 (cid:11)) 2 (cid:0) mainstobeshow (cid:0) nth (cid:0) atV (u;A)agreeswith(26)and(7). Now,since = 1 (cid:21)(cid:11)2 +(cid:18)(1 (cid:11))2 , 1 2 (cid:0) they agree only if (cid:0) (cid:1) 1 = (cid:18)(1 (cid:11)); 1 (cid:12)(cid:11)2 2 (cid:0) (cid:0) i.e., if (cid:21) (cid:11)2 +(1 (cid:11))2 = (1 (cid:11)) 1 (cid:12)(cid:11)2 : (31) (cid:18) (cid:0) (cid:0) (cid:0) (cid:18) (cid:19) But from (28), (cid:0) (cid:1) (cid:21) 1 = +(cid:12)(cid:11) 1 (cid:12): (cid:18) (cid:11) (cid:0) (cid:0) Substitute for (cid:21)=(cid:18) into (31) to conclude that V (u;A) is consistent with (26) and (7) if 1 and only if (cid:11)+(cid:12)(cid:11)3 (cid:11)2 (cid:12)(cid:11)2 +(1 (cid:11))2 = (1 (cid:11)) 1 (cid:12)(cid:11)2 : (32) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) But expanding th(cid:0)e left-hand side of (32) yields (cid:1) (cid:0) (cid:1) (cid:11)+(cid:12)(cid:11)3 (cid:11)2 (cid:12)(cid:11)2 +1+(cid:11)2 2(cid:11) = (cid:12)(cid:11)3 (cid:12)(cid:11)2 +1 (cid:11): (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) Conversely, exp(cid:0)anding the right-hand side of (32) y(cid:1)ields (1 (cid:11)) 1 (cid:12)(cid:11)2 = 1 (cid:11) (cid:12)(cid:11)2 +(cid:12)(cid:11)3: (cid:0) (cid:0) (cid:0) (cid:0) Therefore (32) holds, and V is t(cid:0)herefore g(cid:1)iven by (9). 29

4.2 Proof of Proposition 2 Note that we can re-write 11 as MA(t) process t u = (cid:11)tu +x (cid:11)t s" : t 0 (cid:0) s s=0 X As a result, we have u2 u2 = (u +u )(u u ) t+1 (cid:0) t t+1 t t+1 (cid:0) t t+1 t = (cid:11)t+1 +(cid:11)t u +x (cid:11)t+1 s" + (cid:11)t s" 0 (cid:0) s (cid:0) s ( ( )) s=0 s=0 (cid:0) (cid:1) X X t+1 t (cid:11)t+1 (cid:11)t u +x (cid:11)t+1 s" (cid:11)t s" 0 (cid:0) s (cid:0) s (cid:0) (cid:0) ( ( )) s=0 s=0 (cid:0) (cid:1) X X t = (cid:11)t+1 +(cid:11)t u +x 2 (cid:11)t s" +" (cid:11)t+1 (cid:11)t u +x" : 0 (cid:0) s t+1 0 t+1 (cid:0) ( ( )) s=0 (cid:0) (cid:1) X (cid:8)(cid:0) (cid:16)(cid:1)2 (cid:9) (cid:16) 1 | {z } This is product o|f two variables (cid:16) and{z(cid:16) that follow norma}l distributions 1 2 1 (cid:11)2(t+1) (cid:16) N (cid:11)t+1 +(cid:11)t u ;x2 4 (cid:0) +1 (cid:27)2 1 (cid:24) 0 1 (cid:11)2 (cid:18) (cid:26) (cid:20) (cid:0) (cid:21) (cid:27) (cid:19) (cid:16) N (cid:0)(cid:11)t+1 (cid:11)t (cid:1)u ;x2(cid:27)2 : 2 0 (cid:24) (cid:0) (cid:0)(cid:0) (cid:1) (cid:1) It(cid:146)s worth noting that the product of two variables can be written as 1 1 XY = (X +Y)2 (X Y)2: 4 (cid:0) 4 (cid:0) It follows that since X = (cid:16) and Y = (cid:16) are normal distribution 1 2 1 (cid:11)2(t+1) (cid:16) +(cid:16) 2(cid:11)t+1u N 0;4x2 (cid:0) +1 (cid:27)2 = A(cid:24) 1 2 (cid:0) 0 (cid:24) 1 (cid:11)2 t (cid:18) (cid:20)(cid:20) (cid:0) (cid:21) (cid:21) (cid:19) 1 (cid:11)2(t+1) (cid:16) +(cid:16) 2(cid:11)tu N 0;4x2 (cid:0) (cid:27)2 = B(cid:24) : 1 2 (cid:0) 0 (cid:24) 1 (cid:11)2 t (cid:18) (cid:20) (cid:0) (cid:21) (cid:19) where (cid:24) is a standard normal and t 1 (cid:11)2(t+1) A = 4x2(cid:27)2 (cid:0) +1 1 (cid:11)2 s (cid:20)(cid:20) (cid:0) (cid:21) (cid:21) 1 (cid:11)2(t+1) B = 4x2 (cid:0) (cid:27)2: 1 (cid:11)2 s (cid:20) (cid:0) (cid:21) 30

Therefore, we have 1 1 (cid:16) (cid:16) = A(cid:24) +2(cid:11)t+1u 2 B(cid:24) +2(cid:11)tu 2 1 2 4 t 0 (cid:0) 4 t 0 1 = (cid:0) [A(cid:24) ]2 + 2(cid:11)t+(cid:1)1u 2 + (cid:0) 4(cid:11)t+1u A(cid:24) (cid:1) 4 t 0 0 t (cid:16)1 (cid:17) [B(cid:24) ]2 + (cid:2) 2(cid:11)tu (cid:3)2 +4(cid:11)tu B(cid:24) (cid:0)4 t 0 0 t (cid:16)1 (cid:17) = C + A2 B (cid:2)2 (cid:24)2 + (cid:3) ((cid:11)A B)(cid:11)tu (cid:24) ; 4 (cid:0) t (cid:0) 0 t (cid:2) (cid:3) where the constant C = (cid:11)t+1u 2 (cid:11)tu 2 : 0 0 (cid:0) Now note that (cid:0) (cid:1) (cid:0) (cid:1) A2 B2 = 4x2(cid:27)2; (cid:0) this leads to (cid:16) (cid:16) = C +x2(cid:27)2(cid:24)2 +((cid:11)A B)(cid:11)tu (cid:24) : 1 2 t (cid:0) 0 t We now have the growth distribution g = x C +x2(cid:27)2(cid:24)2 +((cid:11)A B)(cid:11)tu (cid:24) ; t (cid:0) t t (cid:0) 0 t (cid:2) (cid:3) where (cid:24) N (0;1) and A, B and C are as stated in the proposition. t (cid:24) 4.3 Extension with Learning The following proposition provides a closed-form expression for the growth distribution when there is no uncertainty of (cid:23). Proposition 3 When there is no uncertainty of (cid:23) and under log preferences, the optimal policy function and value function satisfy, 1 (cid:12)(cid:27)2 3 (cid:1) = (cid:18) (cid:18) (cid:19) V (u) = B u2; (cid:0) where 2 1 1 (cid:18) (cid:12)(cid:27)2 3 (cid:12)(cid:27)2 (cid:0)3 B = A (cid:12)(cid:27)2 ; 1 (cid:12) (cid:0) 2 (cid:18) (cid:0) (cid:18) ! (cid:0) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21)2 = 2 31

and the log growth g follows 1 (cid:21) (cid:12)(cid:27)2 (cid:0)3 g = (cid:1)lny = u2 (cid:27)2(cid:16)2 : 2 (cid:0) (cid:18) ! (cid:18) (cid:19) Therefore conditional on u, the growth distribution is left skewed 1 (cid:21) (cid:21)(cid:27)2 (cid:12)(cid:27)2 (cid:0)3 E(g) = u2 2 (cid:0) 2 (cid:18) (cid:18) (cid:19) 1 2 (cid:21)(cid:27)2 (cid:12)(cid:27)2 (cid:0)3 (cid:27)2 var(g) = 2 2 (cid:18) (cid:1) ! (cid:18) (cid:19) 1 (cid:21)(cid:27)2p8 (cid:12)(cid:27)2 (cid:0)3 skewness(g) = : (cid:0) 2 (cid:18) (cid:18) (cid:19) Proof. When there is no uncertainty of (cid:23), (cid:28) = 0, (cid:21) (cid:18) (cid:27)2 1=2 V (u) = maxA u2 (cid:1)2 +(cid:12) V (cid:16) dF ((cid:16)) (cid:1) (cid:0) 2 (cid:0) 2 (cid:1) ! Z (cid:18) (cid:19) If we guess and verify V (u) = B u2 and (cid:1) = C (cid:0) (cid:21) (cid:18) (cid:27)2 1=2 V (u) = maxA u2 (cid:1)2 +(cid:12) V (cid:16) dF ((cid:16)) (cid:1) (cid:0) 2 (cid:0) 2 (cid:1) ! Z (cid:18) (cid:19) (cid:21) (cid:18) (cid:27)2 = maxA u2 (cid:1)2 +(cid:12)B (cid:12) (cid:1) (cid:0) 2 (cid:0) 2 (cid:0) (cid:1) (cid:18) (cid:19) First order condition (cid:27)2 (cid:12) = (cid:18)(cid:1) (cid:1)2 (cid:18) (cid:19) or 1 (cid:12)(cid:27)2 3 (cid:1) = (cid:18) (cid:18) (cid:19) For the value function, (cid:21) (cid:18) (cid:27)2 B u2 = A u2 (cid:1)2 +(cid:12)B (cid:12) (cid:0) (cid:0) 2 (cid:0) 2 (cid:0) (cid:1) (cid:18) (cid:19) Matching the coe¢ cients, (cid:18) (cid:27)2 B = A (cid:1)2 +(cid:12)B (cid:12) ; (cid:0) 2 (cid:0) (cid:1) (cid:18) (cid:19) or 2 1 1 (cid:18) (cid:12)(cid:27)2 3 (cid:12)(cid:27)2 (cid:0)3 B = A (cid:12)(cid:27)2 1 (cid:12) (cid:0) 2 (cid:18) (cid:0) (cid:18) ! (cid:0) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21) = ; 2 as stated in the proposition. 32